Deciding universality of quantum gates

We say that collection of n-qudit gates is universal if there exists N0⩾n such that for every N⩾N0 every N-qudit unitary operation can be approximated with arbitrary precision by a circuit built from gates of the collection. Our main result is an upper bound on the smallest N0 with the above property. The bound is roughly d8n, where d is the number of levels of the base system (the ‘d’ in the term qudit). The proof is based on a recent result on invariants of (finite) linear groups.